61.28.11 problem 71

Internal problem ID [12571]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 71
Date solved : Tuesday, January 28, 2025 at 08:02:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x +b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 40

dsolve(x*diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*((a-c)*x+b)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} {\mathrm e}^{-c x}+c_{2} x^{-\frac {b}{2}} \operatorname {WhittakerM}\left (-\frac {b}{2}, -\frac {b}{2}+\frac {1}{2}, \left (a -2 c \right ) x \right ) {\mathrm e}^{-\frac {a x}{2}} \]

✓ Solution by Mathematica

Time used: 0.208 (sec). Leaf size: 50

DSolve[x*D[y[x],{x,2}]+(a*x+b)*D[y[x],x]+c*((a-c)*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-c x} \left (c_1-c_2 x^{1-b} (x (a-2 c))^{b-1} \Gamma (1-b,(a-2 c) x)\right ) \]